In this article, we would illustrate GSM GMSK modulation. Below are the equations specified in 3GPP TS 45.004:2.

Differential encoding is used; it means rather than every bit value, difference in consecutive bits is encoded as +1 or -1 (+1 is when bits are same, -1 otherwise), refer αi below. Along with equations, corresponding waveforms are shown for an example of one differential encoding of "+1".

Gaussian function with BT=0.3, T=48/13 μs

h(t) =

1

δT . √

(2π)

e

-t2

2δ2T2

δ =

____

= 0.44168 with BT=0.3

√ln(2)

2πBT

Convolution

g(t) = h(t) * rect(

t

)

T

rect(

t

) =

1

if |t| <

T

T

T

2

rect(

t

) =

0

if |t| >

T

T

2

Integral

.

t'

.

G(t') = πh .

∫

g(u) du

.

-∞

.

φ(t') =

n

.

t'-iT

.

Σ

( αiπh .

∫

g(u) du )

i=1

.

-∞

.

αi = { +1, -1 }
h = 0.5

Phase shift of carrier

____

x(t') =

√

(

2Ec

T

) . cos(2πf0t' + φ(t') + φ0)

As you notice in above diagram, (modulation) phase shift increases and saturates after 2T. Above plots are drawn with gnuplot, more about it in later article.

In next article, we will take an interesting case of Frequency Correction Burst (FB) transmitted over FCCH.